ONT Re: Probability And Statistics
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PAS. Note 4
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| 1.2. Probability Spaces (cont.)
|
| We now come to the assignment of probabilities to events.
| As was made clear in the examples of the preceding section,
| the probability of an event is a nonnegative real number. For
| an event A, let P(A) denote this number. Then 0 =< P(A) =< 1.
| The set !W! representing every possible outcome should, of course,
| be assigned the number 1, so P(!W!) = 1.
|
| In our discussion of Example 1 we showed that the probability of events
| satisfies the property that if A and B are any two disjoint events then
| P(A |_| B) = P(A) + P(B). Similarly, in Example 2 we showed that if
| A and B are two disjoint intervals, then we should also require that:
|
| P(A |_| B) = P(A) + P(B).
|
| It now seems reasonable in general to demand that if A and B are disjoint
| events then P(A |_| B) = P(A) + P(B). By induction, it would then follow
| that if A_1, A_2, ..., A_n are any n mutually disjoint sets (that is, if
| A_i |^| A_j = {} whenever i =/= j), then:
|
| P(|_| (i = 1 to n) A_i) = Sum (i = 1 to n) P(A_i).
|
| Actually, again for mathematical reasons, we will
| in fact demand that this additivity property hold
| for countable collections of disjoint events.
|
| Definition 2.
|
| A probability measure P on a !s!-field of
| subsets $A$ of a set !W! is a real-valued
| function having domain $A$ satisfying the
| following properties:
|
| 1. P(!W!) = 1.
|
| 2. P(A) >= 0 for all A in $A$.
|
| 3. If A_n, n = 1, 2, 3, ..., are
| mutually disjoint sets in $A$,
| then:
|
| P(|_| (n = 1 to oo) A_n) = Sum (n = 1 to oo) P(A_n).
|
| A probability space, denoted by (!W!, $A$, P),
| is a set !W!, a !s!-field of subsets $A$, and
| a probability measure P defined on $A$.
|
| Hoel, Port, Stone, 'Probability Theory', p. 8.
|
| Hoel, P.G., Port, S.C., & Stone, C.J.,
|'Introduction to Probability Theory',
| Houghton Mifflin, Boston, MA, 1971.
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